On the convergence of the normalized power sequence of spectral operators on Hilbert space
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908707708731392 |
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| author | Nayak, Soumyashant Shekhawat, Renu |
| author_facet | Nayak, Soumyashant Shekhawat, Renu |
| contents | Let $\mathscr{H}$ be a complex Hilbert space, and let $\mathscr{B}(\mathscr{H})$ denote the set of all bounded operators on $\mathscr{H}$ . For an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $II_1$ factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16318 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convergence of the normalized power sequence of spectral operators on Hilbert space Nayak, Soumyashant Shekhawat, Renu Functional Analysis Operator Algebras 47B40, 47A10, 15A18 Let $\mathscr{H}$ be a complex Hilbert space, and let $\mathscr{B}(\mathscr{H})$ denote the set of all bounded operators on $\mathscr{H}$ . For an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $II_1$ factor. |
| title | On the convergence of the normalized power sequence of spectral operators on Hilbert space |
| topic | Functional Analysis Operator Algebras 47B40, 47A10, 15A18 |
| url | https://arxiv.org/abs/2410.16318 |